Template:Regular convex 4-polytopes

class="wikitable mw-collapsible {{{collapsestate|mw-collapsed}}}" style="white-space:nowrap;text-align:center;" !colspan=8|Regular convex 4-polytopes {{#if:{{{radius

}}|of radius {{{radius|}}}|}}

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!style="text-align:right;"|Symmetry group

|A₄

|colspan=2|B₄

|F₄

|colspan=2|H₄

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!style="vertical-align:top;text-align:right;"|Name

|style="vertical-align:top;"|5-cell

Hyper-tetrahedron

5-point

|style="vertical-align:top;"|16-cell

Hyper-octahedron

8-point

|style="vertical-align:top;"|8-cell

Hyper-cube

16-point

|style="vertical-align:top;"|24-cell

24-point

|style="vertical-align:top;"|600-cell

Hyper-icosahedron

120-point

|style="vertical-align:top;"|120-cell

Hyper-dodecahedron

600-point

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!style="text-align:right;"|Schläfli symbol

|{3, 3, 3}

|{3, 3, 4}

|{4, 3, 3}

|{3, 4, 3}

|{3, 3, 5}

|{5, 3, 3}

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!style="text-align:right;"|Coxeter mirrors

|{{Coxeter–Dynkin diagram|node_1|3|node|3|node|3|node}}

|{{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}}

|{{Coxeter–Dynkin diagram|node_1|4|node|3|node|3|node}}

|{{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}

|{{Coxeter–Dynkin diagram|node_1|3|node|3|node|5|node}}

|{{Coxeter–Dynkin diagram|node_1|5|node|3|node|3|node}}

|-

!style="text-align:right;"|Mirror dihedrals

|{{sfrac|𝝅|3}} {{sfrac|𝝅|3}} {{sfrac|𝝅|3}} {{sfrac|𝝅|2}} {{sfrac|𝝅|2}} {{sfrac|𝝅|2}}

|{{sfrac|𝝅|3}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}} {{sfrac|𝝅|2}} {{sfrac|𝝅|2}} {{sfrac|𝝅|2}}

|{{sfrac|𝝅|4}} {{sfrac|𝝅|3}} {{sfrac|𝝅|3}} {{sfrac|𝝅|2}} {{sfrac|𝝅|2}} {{sfrac|𝝅|2}}

|{{sfrac|𝝅|3}} {{sfrac|𝝅|4}} {{sfrac|𝝅|3}} {{sfrac|𝝅|2}} {{sfrac|𝝅|2}} {{sfrac|𝝅|2}}

|{{sfrac|𝝅|3}} {{sfrac|𝝅|3}} {{sfrac|𝝅|5}} {{sfrac|𝝅|2}} {{sfrac|𝝅|2}} {{sfrac|𝝅|2}}

|{{sfrac|𝝅|5}} {{sfrac|𝝅|3}} {{sfrac|𝝅|3}} {{sfrac|𝝅|2}} {{sfrac|𝝅|2}} {{sfrac|𝝅|2}}

|-

!style="vertical-align:top;text-align:right;"|Graph

|120px

|120px

|120px

|120px

|120px

|120px

|-

!style="text-align:right;"|Vertices

|5 tetrahedral

|8 octahedral

|16 tetrahedral

|24 cubical

|120 icosahedral

|600 tetrahedral

|-

!style="vertical-align:top;text-align:right;"|Edges

|10 triangular

|24 square

|32 triangular

|96 triangular

|720 pentagonal

|1200 triangular

|-

!style="vertical-align:top;text-align:right;"|Faces

|10 triangles

|32 triangles

|24 squares

|96 triangles

|1200 triangles

|720 pentagons

|-

!style="vertical-align:top;text-align:right;"|Cells

|5 tetrahedra

|16 tetrahedra

|8 cubes

|24 octahedra

|600 tetrahedra

|120 dodecahedra

|-

!style="vertical-align:top;text-align:right;"|Tori

|1 5-tetrahedron

|2 8-tetrahedron

|2 4-cube

|4 6-octahedron

|20 30-tetrahedron

|12 10-dodecahedron

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!style="vertical-align:top;text-align:right;"|Inscribed

|120 in 120-cell

|675 in 120-cell

|2 16-cells

|3 8-cells

|25 24-cells

|10 600-cells

|-

!style="vertical-align:top;text-align:right;"|Great polygons

|

|2 squares x 3

|4 rectangles x 4

|4 hexagons x 4

|12 decagons x 6

|100 irregular hexagons x 4

|-

!style="vertical-align:top;text-align:right;"|Petrie polygons

|1 pentagon x 2

|1 octagon x 3

|2 octagons x 4

|2 dodecagons x 4

|4 30-gons x 6

|20 30-gons x 4

|-

!style="vertical-align:top;text-align:right;"|{{#ifeq:{{{radius|}}}|1|Long radius|{{#ifeq:{{{radius|}}}|{{radic|2}}|Long radius|Long radius}}}}

|{{#ifeq:{{{radius|1}}}|1|$1$|{{#ifeq:{{{radius}}}|{{radic|2}}|$\backslash sqrt\{2\}$}}}}

|{{#ifeq:{{{radius|1}}}|1|$1$|{{#ifeq:{{{radius}}}|{{radic|2}}|$\backslash sqrt\{2\}$}}}}

|{{#ifeq:{{{radius|1}}}|1|$1$|{{#ifeq:{{{radius}}}|{{radic|2}}|$\backslash sqrt\{2\}$}}}}

|{{#ifeq:{{{radius|1}}}|1|$1$|{{#ifeq:{{{radius}}}|{{radic|2}}|$\backslash sqrt\{2\}$}}}}

|{{#ifeq:{{{radius|1}}}|1|$1$|{{#ifeq:{{{radius}}}|{{radic|2}}|$\backslash sqrt\{2\}$}}}}

|{{#ifeq:{{{radius|1}}}|1|$1$|{{#ifeq:{{{radius}}}|{{radic|2}}|$\backslash sqrt\{2\}$}}}}

|-

!style="vertical-align:top;text-align:right;"|Edge length

|{{#ifeq:{{{radius|1}}}|1|$\backslash sqrt\{\backslash tfrac\{5\}\{2\}\}\; \backslash approx\; 1.581$|{{#ifeq:{{{radius}}}|{{radic|2}}|$\backslash sqrt\{5\}\; \backslash approx\; 2.236$}}}}

|{{#ifeq:{{{radius|1}}}|1|$\backslash sqrt\{2\}\; \backslash approx\; 1.414$|{{#ifeq:{{{radius}}}|{{radic|2}}|$2$}}}}

|{{#ifeq:{{{radius|1}}}|1|$1$|{{#ifeq:{{{radius}}}|{{radic|2}}|$\backslash sqrt\{2\}\; \backslash approx\; 1.414$}}}}

|{{#ifeq:{{{radius|1}}}|1|$1$|{{#ifeq:{{{radius}}}|{{radic|2}}|$\backslash sqrt\{2\}\; \backslash approx\; 1.414$}}}}

|{{#ifeq:{{{radius|1}}}|1|$\backslash tfrac\{1\}\{\backslash phi\}\; \backslash approx\; 0.618$|{{#ifeq:{{{radius}}}|{{radic|2}}|$\backslash tfrac\{\backslash sqrt\{2\}\}\{\backslash phi\}\; \backslash approx\; 0.874$}}}}

|{{#ifeq:{{{radius|1}}}|1|$\backslash tfrac\{1\}\{\backslash phi^2\backslash sqrt\{2\}\}\; \backslash approx\; 0.270$|{{#ifeq:{{{radius}}}|{{radic|2}}|$2-\backslash phi\; \backslash approx\; 0.382$}}}}

|-

!style="vertical-align:top;text-align:right;"|Short radius

|{{#ifeq:{{{radius|1}}}|1|$\backslash tfrac\{1\}\{4\}$|{{#ifeq:{{{radius}}}|{{radic|2}}|$\backslash tfrac\{\backslash sqrt\{2\}\}\{4\}\; \backslash approx\; 0.354$}}}}

|{{#ifeq:{{{radius|1}}}|1|$\backslash tfrac\{1\}\{2\}$|{{#ifeq:{{{radius}}}|{{radic|2}}|$\backslash tfrac\{\backslash sqrt\{2\}\}\{2\}\; \backslash approx\; 0.707$}}}}

|{{#ifeq:{{{radius|1}}}|1|$\backslash tfrac\{1\}\{2\}$|{{#ifeq:{{{radius}}}|{{radic|2}}|$\backslash tfrac\{\backslash sqrt\{2\}\}\{2\}\; \backslash approx\; 0.707$}}}}

|{{#ifeq:{{{radius|1}}}|1|$\backslash sqrt\{\backslash tfrac\{1\}\{2\}\}\; \backslash approx\; 0.707$|{{#ifeq:{{{radius}}}|{{radic|2}}|$1$}}}}

|{{#ifeq:{{{radius|1}}}|1|$\backslash sqrt\{\backslash tfrac\{\backslash phi^4\}\{8\}\}\; \backslash approx\; 0.926$|{{#ifeq:{{{radius}}}|{{radic|2}}|$\backslash sqrt\{\backslash tfrac\{\backslash phi^4\}\{4\}\}\; \backslash approx\; 1.309$}}}}

|{{#ifeq:{{{radius|1}}}|1|$\backslash sqrt\{\backslash tfrac\{\backslash phi^4\}\{8\}\}\; \backslash approx\; 0.926$|{{#ifeq:{{{radius}}}|{{radic|2}}|$\backslash sqrt\{\backslash tfrac\{\backslash phi^4\}\{4\}\}\; \backslash approx\; 1.309$}}}}

|-

!style="vertical-align:top;text-align:right;"|Area

|{{#ifeq:{{{radius|1}}}|1|$10\backslash left(\backslash tfrac\{5\backslash sqrt\{3\}\}\{8\}\backslash right)\; \backslash approx\; 10.825$|{{#ifeq:{{{radius}}}|{{radic|2}}|$10\backslash left(\backslash tfrac\{5\backslash sqrt\{3\}\}\{4\}\backslash right)\; \backslash approx\; 21.651$}}}}

|{{#ifeq:{{{radius|1}}}|1|$32\backslash left(\backslash sqrt\{\backslash tfrac\{3\}\{4\}\}\backslash right)\; \backslash approx\; 27.713$|{{#ifeq:{{{radius}}}|{{radic|2}}|$32\backslash left(\backslash sqrt\{3\}\backslash right)\; \backslash approx\; 55.425$}}}}

|{{#ifeq:{{{radius|1}}}|1|$24$|{{#ifeq:{{{radius}}}|{{radic|2}}|$48$}}}}

|{{#ifeq:{{{radius|1}}}|1|$96\backslash left(\backslash sqrt\{\backslash tfrac\{3\}\{16\}\}\backslash right)\; \backslash approx\; 41.569$|{{#ifeq:{{{radius}}}|{{radic|2}}|$96\backslash left(\backslash sqrt\{\backslash tfrac\{3\}\{4\}\}\backslash right)\; \backslash approx\; 83.138$}}}}

|{{#ifeq:{{{radius|1}}}|1|$1200\backslash left(\backslash tfrac\{\backslash sqrt\{3\}\}\{4\backslash phi^2\}\backslash right)\; \backslash approx\; 198.48$|{{#ifeq:{{{radius}}}|{{radic|2}}|$1200\backslash left(\backslash tfrac\{2\backslash sqrt\{3\}\}\{4\backslash phi^2\}\backslash right)\; \backslash approx\; 396.95$}}}}

|{{#ifeq:{{{radius|1}}}|1|$720\backslash left(\backslash tfrac\{\backslash sqrt\{25+10\backslash sqrt\{5\}\}\}\{8\backslash phi^4\}\backslash right)\; \backslash approx\; 90.366$|{{#ifeq:{{{radius}}}|{{radic|2}}|$720\backslash left(\backslash tfrac\{\backslash sqrt\{25+10\backslash sqrt\{5\}\}\}\{4\backslash phi^4\}\backslash right)\; \backslash approx\; 180.73$}}}}

|-

!style="vertical-align:top;text-align:right;"|Volume

|{{#ifeq:{{{radius|1}}}|1|$5\backslash left(\backslash tfrac\{5\backslash sqrt\{5\}\}\{24\}\backslash right)\; \backslash approx\; 2.329$|{{#ifeq:{{{radius}}}|{{radic|2}}|$5\backslash left(\backslash tfrac\{5\backslash sqrt\{10\}\}\{12\}\backslash right)\; \backslash approx\; 6.588$}}}}

|{{#ifeq:{{{radius|1}}}|1|$16\backslash left(\backslash tfrac\{1\}\{3\}\backslash right)\; \backslash approx\; 5.333$|{{#ifeq:{{{radius}}}|{{radic|2}}|$16\backslash left(\backslash tfrac\{2\backslash sqrt\{2\}\}\{3\}\backslash right)\; \backslash approx\; 15.085$}}}}

|{{#ifeq:{{{radius|1}}}|1|$8$|{{#ifeq:{{{radius}}}|{{radic|2}}|$8\backslash sqrt\{8\}\; \backslash approx\; 22.627$}}}}

|{{#ifeq:{{{radius|1}}}|1|$24\backslash left(\backslash tfrac\{\backslash sqrt\{2\}\}\{3\}\backslash right)\; \backslash approx\; 11.314$|{{#ifeq:{{{radius}}}|{{radic|2}}|$24\backslash left(\backslash tfrac\{4\}\{3\}\backslash right)\; =\; 32$}}}}

|{{#ifeq:{{{radius|1}}}|1|$600\backslash left(\backslash tfrac\{\backslash sqrt\{2\}\}\{12\backslash phi^3\}\backslash right)\; \backslash approx\; 16.693$|{{#ifeq:{{{radius}}}|{{radic|2}}|$600\backslash left(\backslash tfrac\{4\}\{12\backslash phi^3\}\backslash right)\; \backslash approx\; 47.214$}}}}

|{{#ifeq:{{{radius|1}}}|1|$120\backslash left(\backslash tfrac\{15\; +\; 7\backslash sqrt\{5\}\}\{4\backslash phi^6\backslash sqrt\{8\}\}\backslash right)\; \backslash approx\; 18.118$|{{#ifeq:{{{radius}}}|{{radic|2}}|$120\backslash left(\backslash tfrac\{15\; +\; 7\backslash sqrt\{5\}\}\{4\backslash phi^6\}\backslash right)\; \backslash approx\; 51.246$}}}}

|-

!style="vertical-align:top;text-align:right;"|4-Content

|{{#ifeq:{{{radius|1}}}|1|$\backslash tfrac\{\backslash sqrt\{5\}\}\{24\}\backslash left(\backslash tfrac\{\backslash sqrt\{5\}\}\{2\}\backslash right)^4\; \backslash approx\; 0.146$|{{#ifeq:{{{radius}}}|{{radic|2}}|$\backslash tfrac\{\backslash sqrt\{5\}\}\{24\}\backslash left(\backslash sqrt\{5\}\backslash right)^4\; \backslash approx\; 2.329$}}}}

|{{#ifeq:{{{radius|1}}}|1|$\backslash tfrac\{2\}\{3\}\; \backslash approx\; 0.667$|{{#ifeq:{{{radius}}}|{{radic|2}}|$\backslash tfrac\{8\}\{3\}\; \backslash approx\; 2.666$}}}}

|{{#ifeq:{{{radius|1}}}|1|$1$|{{#ifeq:{{{radius}}}|{{radic|2}}|$4$}}}}

|{{#ifeq:{{{radius|1}}}|1|$2$|{{#ifeq:{{{radius}}}|{{radic|2}}|$8$}}}}

|{{#ifeq:{{{radius|1}}}|1|$\backslash tfrac\{\backslash text\{Short\}\backslash times\backslash text\{Vol\}\}\{4\}\; \backslash approx\; 3.863$|{{#ifeq:{{{radius}}}|{{radic|2}}|$\backslash tfrac\{\backslash text\{Short\}\backslash times\backslash text\{Vol\}\}\{4\}\; \backslash approx\; 15.451$}}}}

|{{#ifeq:{{{radius|1}}}|1|$\backslash tfrac\{\backslash text\{Short\}\backslash times\backslash text\{Vol\}\}\{4\}\; \backslash approx\; 4.193$|{{#ifeq:{{{radius}}}|{{radic|2}}|$\backslash tfrac\{\backslash text\{Short\}\backslash times\backslash text\{Vol\}\}\{4\}\; \backslash approx\; 16.770$}}}}

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Category:Geometry templates